Explanation:
(A) : Given equation of lines are,
\(x \sin \theta+y \cos \theta=5 \cos 2 \theta\)
\(\cos x \operatorname{cosec} \theta+y \sec \theta=5\)
Given that \(P\), is perpendicular distance from origin \((0,0)\) to line(1)
\(P_{1}=\frac{|\sin \theta \times(0)+\cos \theta \times 0-5 \cos 2 \theta|}{\sqrt{(\sin \theta)^{2}+(\cos \theta)^{2}}} \tag{iii}\)
\(\mathrm{P}=5 \cos 2 \theta\)
And, \(\mathrm{P}_{2}\) is perpendicular distance from origin \((0,0)\) to line (ii)
Now,
\(P_{2} =\frac{|\operatorname{cosec} \theta \times 0+\sec \theta \times 0-5|}{\sqrt{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}}\)
\(=\frac{5}{\sqrt{\operatorname{cosec}^{2} \theta}+\sec ^{2} \theta} \tag{iv}\)
\(P_{1}^{2}+4 P_{2}^{2}=(5 \cos 2 \theta)^{2}+4\left(\frac{5}{\sqrt{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}}\right)^{2}\)
\(=25 \cos ^{2} 2 \theta+\frac{4 \times 25}{\operatorname{cosec} e^{2} \theta+\sec ^{2} \theta}\)
\(=25 \cos ^{2} 2 \theta+\frac{100}{\left(\frac{1}{\sin ^{2}}\right)+\left(\frac{1}{\cos ^{2} \theta}\right)}\)
\(=25 \cos ^{2} 2 \theta+100\left(\sin ^{2} \theta \cdot \operatorname{Cos}^{2} \theta\right)\)
\(=25\left[\cos ^{2} 2 \theta+4 \sin ^{2} \theta \cos ^{2} \theta\right]=25\left[\cos ^{2} 2 \theta+(2 \sin \theta \cos \theta)^{2}\right]\)
\(=25\left[\cos ^{2} 2 \theta+\sin ^{2} 2 \theta\right]=25 \times[1]=25\)
